Arsenal loanee Aneke and livewire Mesca along with Brad Inman proved the difference between the two sides as the Railwaymen registered back-to-back wins for the first time this season.
Aneke's mastery in the air almost put Crewe ahead in the 15th minute as he rose the highest to Matt Tootle's cross, only for his looping header to be tipped on to the crossbar by Ben Amos.
Crewe were more inventive and more determined. Inman was inches away from opening the scoring when he raced clear down the left on the stroke of half-time. He held off veteran full-back Pascal Chimbonda, but arrowed a shot the wrong side of the far post.
But David Amoo went close to putting the Cumbrians ahead with a lovely piece of skill from the right of the box. The winger's curling left-footed effort beat goalkeeper Ben Garratt, but bounced back off the inside of the post.
Crewe made the most of their good fortune to forge in front in the 52nd minute with Aneke unleashing a chipped finish from the right of a congested box and the ball dropping over Amos into the far corner.
The home side then stepped up another gear when Inman broke clear to expose Carlisle's problems on the right flank. The winger's cross deflected conveniently into the path of Mesca, who made no mistake with his close-range finish in the 63rd minute.
Carlisle responded strongly and Liam Noble went close with a 25-yard drive which just cleared the crossbar.
Lawrence reduced the arrears with a super free-kick which flew into the corner from 25 yards out with 14 minutes left before Crewe survived a late onslaught which saw Kelvin Mellor make a goal-line clearance.